The Random Walk: What Simple Probability Suggest about Society

Imagine you set out on a special “walk”. Your journey’s direction is to be dictated by a friend flipping a coin. You either walk forward or backward a step depending on whether they flipped heads or tails.

If you were to guess your final position relative to your starting point after 10 coin flips, where would you guess?

The most probable guess is where you started, and yet the chance that you got exactly 5 heads and 5 tails (the outcome needed for this final position) is not guaranteed by any means. Rather….you are likely “close” to your starting position, as the many “paths” of achieving “close”: 4 tails 6 heads, 5 tails 5 heads, 6 tails 4 heads, etc are more numerous relative to the few “paths” to fringe outcomes (9 tails 1 head, 1 tail 9 heads, etc).

This is visualized in the Galton board, where marbles hitting pegs and going left or right is kind of like coins being tossed and landing heads or tails:

“Galton Board”

Image Credit from Alan Zucconi

The outcome is a normal distribution centered around equal left/right or heads/tails.

Yet…..as we take more steps (flip more coins)... there is also more POTENTIAL to have  traveled further from our starting point, as each step could potentially lead further and further.
And so despite the outcomes still centering on your starting point ... .the probability of “drifting” away from your starting point is higher.

Image credit: Feynman Lectures: Volume I

The figure above shows this increased chance of drifting with more steps (taken from Feynman lectures).

This notion and probabilities can be calculate through the following “Gaussian” or normal distribution expounded on below

(via excerpt from Feynman Lectures: Volume I , Chapter 6: Probability)

So what could this mean for us outside of coin tosses and simpler quantitative applications in natural sciences? Could there be implications for social sciences?

Perhaps the random walk could mean that even in sociocultural “still air” — absent of deliberate currents and forces influencing the complex actions and “motions”  of you and others throughout life — raw probability dictates that with time and increased volume of people/actions there will be “diffusion” and wider distribution of outcomes.

Soft sciences are so complex that they do not easily invite mathematical analysis….but the general intuition of the phenomena/probability of the random walk stands to reason as a contribution to different changes seen in many dynamic systems.

Some systems theorist will invoke the attractor states, causality, and other systemic dispositions as challenges to this proposition, but I think there is room for this loose concept in the inevitable stochastic fluctuations inherent to areas of life.

Such examples of social attractor states could be our biological tendencies, sociocultural moors/conventions, technology, and normalization processes such as school/organized religion, which very well may mitigate deviation ….. .but even allegedly static attractors/ institutions could be subject to subtle, or even insidious, drift.

The result? As societies age and grow…the potential to deviate expands and the ability to predict any single outcome shrinks.